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Chapter 31: Problem 28

You're writing specifications for a new line of magnifying glasses that have double-convex lenses with equal 32 -cm curvature radii, made from glass with \(n=1.52 .\) What do you list for the focal length?

Short Answer

Expert verified
The focal length of the magnifying glasses should be listed as approximately 48 cm.

Step by step solution

01

Recall the Lensmaker's Equation

For a lens in air, the lensmaker's equation is given by \(\frac{1}{f}=(n-1) \left(\frac{1}{R_1} - \frac{1}{R_2} \right)\), where \(f\) is the focal length, \(n\) is the refractive index, and \(R_1\) and \(R_2\) are the radii of curvature for the two sides of the lens. In the case of a double-convex lens, \(R_1\) and \(R_2\) are equal and they are both positive as they are measured from the corresponding centres of curvatures to the lens pointing away from the lens. Since the curvature radius is the same on both sides, \(R_1 = R_2 = 32\) cm.
02

Apply Values to the Lensmaker's Equation

Substitute the given values into the equation. The refractive index \(n=1.52\). Because it's a double-convex lens, both radii have the same magnitude but opposite signs, \(\frac{1}{R_1} - \frac{1}{R_2}= \frac{1}{32 cm} - \frac{1}{-32 cm} = \frac{2}{32 cm} = \frac{1}{16 cm}\). This reduces the lens maker's equation to \(\frac{1}{f}=(1.52-1) \left(\frac{1}{16 cm} \right)\).
03

Solve for the Focal Length

Solve the equation for \(f\). After calculation, one obtains the focal length as \(f = 1/[(1.52-1) * (1/16 cm^{-1})] \approx 48 cm\).

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